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Full text of "Men Of Mathematics"

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MEN OF MATHEMATICS Monsieur Gauss [Gauss was still living when Hermite wrote this, hence the polite 'Monsieur'] has opened up to us, Algebra and the Theory of Numbers seem necessarily to be merged in the same order of analytical concepts, of which our present know- ledge does not yet permit us to form an accurate idea.* He then makes a remark which, although not very clear, can be interpreted as meaning that the key to the subtle connexions between algebra, the higher arithmetic, and certain parts of the theory of functions will be found in a thorough understanding of what sort of 'numbers' are both necessary and sufficient for the explicit solution of all types of algebraic equations. Thus, for a?5 — 1 = 0, it is necessary and sufficient to understand tf 1; for xs + ax + b = 0, where <z,5 are any given numbers, what sort of a 'number' x must be invented in order that % may be expressed explicitly in terms of a,b? Gauss of course gave one kind of answer: any root x is a complex number* But this is only a beginning. Abel proved that if only a finite number of rational operations and extractions of roots are permitted, then there is fzo explicit formula giving x in terms of a,6. We shall return to this question later; Hermite even at this early date (184-8; he was then twenty-six) seems' to have had one of his greatest discoveries somewhere at the back of his head. In his attitude toward numbers Hermite was somewhat of a mystic in the tradition of Pythagoras and Descartes - the latter's mathematical creed, as will appear in a moment, was essentially Pythagorean. In other matters, too, the gentle Hermite exhibited a marked leaning toward mysticism. Up to the age of forty-three he was a tolerant agnostic, like so many French men of science of his time. Then, in 1856, he fell suddenly and dangerously ill. In this debilitated condition he was no match for even the least persistent evangelist, and the ardent Cauchy, who had always deplored his brilliant young friend's open-mindedness on religious matters, pounced on the prostrate Hermite and converted him to Roman Catholicism. Thence- forth Hermite was a devout Catholic, and the practice of his religion gave him much satisfaction. Hermite's number-mysticism is harmless enough and it is one of those personal things on which argument is futfle. Briefly, 504